What:
Every 35-40 seconds, the network stops working and a message pops up at the bottom of my screen with "Connection failed Activation of network connection failed." This message appears for about 5 seconds and then disappears; the network reconnects. This repeats indefinitely.

When:
Started with yum update which upgraded kernel to 3.3.0-4... problem persists with kernel 3.3.0-8. The previous known working kernel is 3.2.10-3.

Hardware:
HP z600, connected to network via gigE. The network is fine; everything works when I boot with 3.2.10-3.

Naturally, I'd like to keep my kernel up to date, but I simply cannot tolerate a broken network driver. What do I do?

There are plenty of reports of similar issues on redhat's bug tracker. From crawling around their site, it seems that it's an IPv6 related issue. As such, disabling IPv6 support fixes this (tested only on latest kernel 3.3.1, which was otherwise still broken).